Aside from squeaks and loose railings, there's not much that can go wrong with a staircase. Like floors, stairs squeak when the lumber becomes warped or when loose boards rub against nails or each other. Eliminating squeaks is easiest if you have access to the underside of the staircase. If not, you can work on noisy steps from above, provided there's no carpet in the way.
Loose railings are shabby and unsafe. With freestanding railings, which are supported by solid or hollow newel posts at the ends of the staircase and at the landings, the problem may lie in the newels' structural connections or in the railing's connection with the newels. Wall-mounted railings are supported by brackets with screws driven into wall studs. If these come loose, tighten the screws, or install longer screws to increase their holding power.
Balusters, the vertical spindles that stand between the steps and the railing, do little to prevent horizontal sway, but a broken baluster is unattractive and can be dangerous, especially for children. You can replace a baluster with a stock piece from a lumber yard or have a wood-turner create a custom duplicate for you.